Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv problms with thm. In Chaptr w larnd about th drivativ or unctions o two variabls. Drivativs told us about th shap o th unction and lt us ind local ma and min w want to b abl to do th sam thing with a unction o two variabls. First lt s think. Imagin a surac th graph o a unction o two variabls. Imagin that th surac is smooth and has som hills and som valls. Concntrat on on point on our surac. What do w want th drivativ to tll us? It ought to tll us how quickl th hight o th surac changs as w mov. Wait which dirction do w want to mov? This is th rason that drivativs ar mor complicatd or unctions o svral variabls thr ar so man dirctions w could mov rom an point. It turns out that our ida o iing on variabl and watching what happns to th unction as th othr changs is th k to tnding th ida o drivativs to mor than on variabl. Partial Drivativs Partial Drivativs: Suppos that z = ( ) is a unction o two variabls. Th partial drivativ o with rspct to is th drivativ o th unction () whr w think o as th onl variabl and act as i is a constant. Th partial drivativ o with rspct to is th drivativ o th unction () whr w think o as th onl variabl and act as i is a constant. Th with rspct to or with rspct to part is rall important ou hav to know and tll which variabl ou ar thinking o as THE variabl. Gomtricall th partial drivativ with rspct to givs th slop o th curv as ou travl along a cross-sction a curv on th surac paralll to th -ais. Th partial drivativ with rspct to givs th slop o th cross-sction paralll to th -ais. Notation or th Partial Drivativ: Th partial drivativ o = () with rspct to is writtn as or z simpl z Th Libniz notation is or d d z W us an adaptation o th notation to man ind th partial drivativ o () with d rspct to : This chaptr is (c) 01. It was rmid b David Lippman rom Shana Calawa's rmi o Contmporar Calculus b Dal Homan. It is licnsd undr th Crativ Commons Attribution licns.
Chaptr Functions o Two Variabls Applid Calculus 6 To stimat a partial drivativ rom a tabl or contour diagram: Th partial drivativ with rspct to can b approimatd b looking at an avrag rat o chang or th slop o a scant lin ovr a vr tin intrval in th -dirction (holding constant). Th tinir th intrval th closr this is to th tru partial drivativ. To comput a partial drivativ rom a ormula: I () is givn as a ormula ou can ind th partial drivativ with rspct to algbraicall b taking th ordinar drivativ thinking o as th onl variabl (holding id). O cours vrthing hr works th sam wa i w r tring to ind th partial drivativ with rspct to just think o as our onl variabl and act as i is constant. Th ida o a partial drivativ works prctl wll or a unction o svral variabls ou ocus on on variabl to b THE variabl and act as i all th othr variabls ar constants. Eampl 1 Hr is a contour diagram or a unction g(). Us th diagram to answr th ollowing qustions: g a. Estimat g and b. Whr on this diagram is g gratst? Whr is g gratst? g mans w'r thinking o as th onl variabl so w ll hold id at =. That mans w ll b looking along th horizontal lin =. To stimat g w nd two unction valus. ( ) lis on th contour lin so w know that g( ) = 0.6. Th nt point as w mov to th right is g(.) = 0.7. a. Now w can ind th avrag rat o chang: g 0.7 0.6 1 Avrag rat o chang = (chang in output) / (chang in input). 08.. 1 W can do th sam thing b going to th nt point w can rad to th lt which is g(.) = g 0. 0.6 1 0.. Thn th avrag rat o chang is. 167.. 6 g givn th inormation w hav or ou could Eithr o ths would b a in stimat o tak thir avrag. W can stimat that. 1 g.
Chaptr Functions o Two Variabls Applid Calculus 6 Estimat g th sam wa but moving on th vrtical lin. Using th nt point up w g 0.7 0.6 gt th avrag rat o chang. 1. Using th nt point down w gt.8 g 0. 0.6.. Taking thir avrag w stimat g. 16.. b. g mans is m onl variabl and w'r thinking o as a constant. So w'r thinking about moving across th diagram on horizontal lins. g will b gratst whn th contour lins g ar closst togthr whn th surac is stpst thn th dnominator in will b small so g will b big. Scanning th graph w can s that th contour lins ar closst togthr whn w had to th lt or to th right rom about (0. 8) and (9 8). So 8) and (9 8). For g I want to look at vrtical lins. 1). g is gratst at about (0. g is gratst at about (.8) and ( Eampl Cold tmpraturs l coldr whn th wind is blowing. Windchill is th prcivd tmpratur and it dpnds on both th actual tmpratur and th wind spd a unction o two variabls! You can rad mor about windchill at http://www.nws.noaa.gov/om/windchill/. Blow is a tabl (courts o th National Wathr Srvic) that shows th prcivd tmpratur or various tmpraturs and windspds. Not that th also includ th ormula but or this ampl w'll us th inormation in th tabl. a. What is th prcivd tmpratur whn th actual tmpratur is F and th wind is blowing at 1 mils pr hour? b. Suppos th actual tmpratur is F. Us inormation rom th tabl to dscrib how th prcivd tmpratur would chang i th wind spd incrasd rom 1 mils pr hour?
Chaptr Functions o Two Variabls Applid Calculus 6 a. Rading th tabl w s that th prcivd tmpratur is 1 F. b. This is a qustion about a partial drivativ. W r holding th tmpratur (T) id at F and asking what happns as wind spd (V) incrass rom 1 mils pr hour. W r thinking o V as th onl variabl so w want WindChill V = W V whn T = and V = 1. W ll ind th avrag rat o chang b looking in th column whr T = and ltting V incras and us that to approimat th partial drivativ. W 11 1 W V 0. V 0 1 What ar th units? W is masurd in F and V is masurd in mph so th units hr ar F/mph. And that lts us dscrib what happns: Th prcivd tmpratur would dcras b about. F or ach mph incras in wind spd. Eampl Find and at th points (0 0) and (1 1) i To ind tak th ordinar drivativ o with rspct to acting as i is constant: Not that th drivativ o th trm with rspct to is zro it s a constant. Similarl 8. Now w can valuat ths at th points:
Chaptr Functions o Two Variabls Applid Calculus 6 00 0 and 00 0 both lat at (00). 1 1 and 1 1 ; this tlls us that th cross sctions paralll to th - and - as ar ; this tlls us that abov th point (1 1) th surac dcrass i ou mov to mor positiv valus and incrass i ou mov to mor positiv valus. Eampl Find and i ln mans is our onl variabl w r thinking o as a constant. Thn w ll just ind th ordinar drivativ. From s point o viw this is an ponntial unction dividd b a constant with a constant addd. Th constant pulls out in ront th drivativ o th ponntial unction is th sam thing and w nd to us th chain rul so w multipl b th drivativ o that ponnt (which is just 1): 1 mans that w r thinking o as th variabl acting as i is constant. From s point o viw is a quotint plus a product w ll nd th quotint rul and th product rul: 1 1 1 1 ln Eampl Find i z z w w z z 1 z mans w act as i z is our onl variabl so w ll act as i all th othr variabls ( and w) ar constants and tak th ordinar drivativ. 1 z z w z z
Chaptr Functions o Two Variabls Applid Calculus 66 Using Partial Drivativs to Estimat Function Valus W can us th partial drivativs to stimat valus o a unction. Th gomtr is similar to th tangnt lin approimation in on variabl. Rcall th on-variabl cas: i is clos nough to a known point a thn a ' a a. In two variabls w do th sam thing in both dirctions at onc: Approimating Function Valus with Partial Drivativs To approimat th valu o ( ) ind som point (a b) whr 1. ( ) and (a b) ar clos that is and a ar clos and and b ar clos.. You know th act valus o (a b) and both partial drivativs thr. Thn a b a b a a b b Notic that th total chang in is bing approimatd b adding th approimat changs coming rom th and dirctions. Anothr wa to look at th sam ormula: How clos is clos? It dpnds on th shap o th graph o. In gnral th closr th bttr. Eampl 6 Us partial drivativs to stimat th valu o at (0.9 1.1) Not that th point (0.9 1.1) is clos to an as point (1 1). In act w alrad workd out th partial drivativs at (1 1): ; 1 1. 8 ; 11. W also know that 1 1 1. So 0.91.1 1 0.1 0.1 1.6. Not that it would hav bn possibl in this cas to simpl comput th act answr; 0.91.1 0.9 0.9 1.1 1.1 1. 69. Our stimat is not prct but it s prtt clos.
Chaptr Functions o Two Variabls Applid Calculus 67 Eampl 7 Hr is a contour diagram or a unction g(). Us partial drivativs to stimat th valu o g(..7). This is th sam diagram rom bor so w alrad stimatd th valu o th unction and th partial drivativs at th narb point (). g( ) is 0.6 our stimat o g. 1 and our stimat o g. 16. So..7 0.6.1..16..76. g Not that in this cas w hav no wa to know how clos our stimat is to th actual valu.
Chaptr Functions o Two Variabls Applid Calculus 68. Erciss For problms 1 through 16 ind and or th unction givn 1... 6.. 1 6. 7. 6 8. 6 ln 9. 10. 11. 1. 16 1 1. 7 1. 6 1. 16.
Chaptr Functions o Two Variabls Applid Calculus 69 17. Hr is a tabl showing th unction A t r t r.0.0.0.06.07 1 0. 0.81 1.7 61.8 7.1 61.8 8.9 10.17 17.0 10.7 9.17 17.0 161.8 197..68 a. Estimat A t.0. b. Estimat A r.0 c. Us our answrs to parts a and b to stimat th valu o A..0 which shows th intrst arnd i 1000 dollars is dpositd in an account arning r annual intrst compoundd continuousl and lt thr or t ars. How clos ar our stimats rom parts a b and c? rt d. Th valus in th tabl cam rom A t r 1000 1 18. Hr is a tabl showing valus or th unction t h H. 100 10 00 t h 0 100 10 00 1 110.1 160.1 10.1 110. 160. 10. 100.9 10.9 00.9 81.6 11.6 181.6. 10. 1. H a. Estimat th valu o dt at ( 10). H b. Estimat th valu o dh at ( 10). c. Us our answrs to parts a and b to stimat th valu o H.616. d. Th valus in th tabl cam rom Ht h h 1t.9t which givs th hight in mtrs abov th ground atr t sconds o an objct that is thrown upward rom an initial hight o h mtrs with an initial vlocit o 1 mtrs pr scond. How clos ar our stimats rom parts a b and c? 19. Givn th unction a. Calculat and b. Us our answrs rom part a to stimat 1.9.1 0. Givn th unction ln 10 and b. Us our answrs rom part a to stimat 1.8.8 a. Calculat
Chaptr Functions o Two Variabls Applid Calculus 70 In problms 1-6 us th contour plot shown to stimat th dsird valu. 1. 1... 00. 1 6.